Find The Value Of Each Given Trigonometric Function Using The Pythagorean Identity.Find Cos ⁡ Θ \cos \theta Cos Θ If 0 ∘ \textless Θ \textless 90 ∘ 0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ} 0 ∘ \textless Θ \textless 9 0 ∘ And Sin ⁡ Θ = 3 2 \sin \theta = \frac{\sqrt{3}}{2} Sin Θ = 2 3 ​ ​ .$\cos \theta

Introduction

The Pythagorean identity is a fundamental concept in trigonometry that relates the sine, cosine, and tangent functions. It is a powerful tool for solving trigonometric equations and finding the values of trigonometric functions. In this article, we will explore how to use the Pythagorean identity to find the value of each given trigonometric function.

The Pythagorean Identity

The Pythagorean identity is given by:

$$\sin^2 \theta + \cos^2 \theta = 1$$

This identity is a fundamental concept in trigonometry and is used to relate the sine and cosine functions. It can be used to find the value of one trigonometric function if the value of the other function is known.

Finding $\cos \theta$ Using the Pythagorean Identity

We are given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$. We want to find the value of $\cos \theta$ using the Pythagorean identity.

First, we can square both sides of the equation $\sin \theta = \frac{\sqrt{3}}{2}$ to get:

$$\sin^2 \theta = \left(\frac{\sqrt{3}}{2}\right)^2$$

Simplifying the right-hand side, we get:

$$\sin^2 \theta = \frac{3}{4}$$

Now, we can substitute this value into the Pythagorean identity:

$$\frac{3}{4} + \cos^2 \theta = 1$$

Subtracting $\frac{3}{4}$ from both sides, we get:

$$\cos^2 \theta = \frac{1}{4}$$

Taking the square root of both sides, we get:

$$\cos \theta = \pm \frac{1}{2}$$

However, since $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$, we know that $\cos \theta$ is positive. Therefore, we can conclude that:

$$\cos \theta = \frac{1}{2}$$

Conclusion

In this article, we used the Pythagorean identity to find the value of $\cos \theta$ given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$. We showed that $\cos \theta = \frac{1}{2}$.

The Pythagorean Identity in Trigonometry

The Pythagorean identity is a fundamental concept in trigonometry that relates the sine, cosine, and tangent functions. It is a powerful tool for solving trigonometric equations and finding the values of trigonometric functions.

The Pythagorean Identity Formula

The Pythagorean identity formula is given by:

$$\sin^2 \theta + \cos^2 \theta = 1$$

This formula can be used to find the value of one trigonometric function if the value of the other function is known.

Using the Pythagorean Identity to Find $\cos \theta$

We can use the Pythagorean identity to find the value of $\cos \theta$ given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$.

First, we can square both sides of the equation $\sin \theta = \frac{\sqrt{3}}{2}$ to get:

$$\sin^2 \theta = \left(\frac{\sqrt{3}}{2}\right)^2$$

Simplifying the right-hand side, we get:

$$\sin^2 \theta = \frac{3}{4}$$

Now, we can substitute this value into the Pythagorean identity:

$$\frac{3}{4} + \cos^2 \theta = 1$$

Subtracting $\frac{3}{4}$ from both sides, we get:

$$\cos^2 \theta = \frac{1}{4}$$

Taking the square root of both sides, we get:

$$\cos \theta = \pm \frac{1}{2}$$

However, since $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$, we know that $\cos \theta$ is positive. Therefore, we can conclude that:

$$\cos \theta = \frac{1}{2}$$

The Importance of the Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry that has many important applications. It is used to solve trigonometric equations, find the values of trigonometric functions, and relate the sine, cosine, and tangent functions.

Using the Pythagorean Identity to Solve Trigonometric Equations

We can use the Pythagorean identity to solve trigonometric equations. For example, if we are given the equation $\sin \theta = \frac{\sqrt{3}}{2}$, we can use the Pythagorean identity to find the value of $\cos \theta$.

First, we can square both sides of the equation $\sin \theta = \frac{\sqrt{3}}{2}$ to get:

$$\sin^2 \theta = \left(\frac{\sqrt{3}}{2}\right)^2$$

Simplifying the right-hand side, we get:

$$\sin^2 \theta = \frac{3}{4}$$

Now, we can substitute this value into the Pythagorean identity:

$$\frac{3}{4} + \cos^2 \theta = 1$$

Subtracting $\frac{3}{4}$ from both sides, we get:

$$\cos^2 \theta = \frac{1}{4}$$

Taking the square root of both sides, we get:

$$\cos \theta = \pm \frac{1}{2}$$

However, since $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$, we know that $\cos \theta$ is positive. Therefore, we can conclude that:

$$\cos \theta = \frac{1}{2}$$

Conclusion

In this article, we used the Pythagorean identity to find the value of $\cos \theta$ given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$. We showed that $\cos \theta = \frac{1}{2}$.

Final Thoughts

The Pythagorean identity is a fundamental concept in trigonometry that has many important applications. It is used to solve trigonometric equations, find the values of trigonometric functions, and relate the sine, cosine, and tangent functions. We hope that this article has provided a clear understanding of the Pythagorean identity and its applications.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometry for Dummies" by Mary Jane Sterling

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry
• [3] Wolfram Alpha: Trigonometry
  

Introduction

In our previous article, we explored how to use the Pythagorean identity to find the value of each given trigonometric function. In this article, we will answer some of the most frequently asked questions about the Pythagorean identity and its applications.

Q&A

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that relates the sine, cosine, and tangent functions. It is given by:

$$\sin^2 \theta + \cos^2 \theta = 1$$

Q: How is the Pythagorean identity used in trigonometry?

A: The Pythagorean identity is used to solve trigonometric equations, find the values of trigonometric functions, and relate the sine, cosine, and tangent functions.

Q: Can the Pythagorean identity be used to find the value of $\cos \theta$ given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$?

A: Yes, we can use the Pythagorean identity to find the value of $\cos \theta$ given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$. First, we can square both sides of the equation $\sin \theta = \frac{\sqrt{3}}{2}$ to get:

$$\sin^2 \theta = \left(\frac{\sqrt{3}}{2}\right)^2$$

Simplifying the right-hand side, we get:

$$\sin^2 \theta = \frac{3}{4}$$

Now, we can substitute this value into the Pythagorean identity:

$$\frac{3}{4} + \cos^2 \theta = 1$$

Subtracting $\frac{3}{4}$ from both sides, we get:

$$\cos^2 \theta = \frac{1}{4}$$

Taking the square root of both sides, we get:

$$\cos \theta = \pm \frac{1}{2}$$

However, since $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$, we know that $\cos \theta$ is positive. Therefore, we can conclude that:

$$\cos \theta = \frac{1}{2}$$

Q: Can the Pythagorean identity be used to find the value of $\sin \theta$ given that $\cos \theta = \frac{1}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$?

A: Yes, we can use the Pythagorean identity to find the value of $\sin \theta$ given that $\cos \theta = \frac{1}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$. First, we can square both sides of the equation $\cos \theta = \frac{1}{2}$ to get:

$$\cos^2 \theta = \left(\frac{1}{2}\right)^2$$

Simplifying the right-hand side, we get:

$$\cos^2 \theta = \frac{1}{4}$$

Now, we can substitute this value into the Pythagorean identity:

$$\sin^2 \theta + \frac{1}{4} = 1$$

Subtracting $\frac{1}{4}$ from both sides, we get:

$$\sin^2 \theta = \frac{3}{4}$$

Taking the square root of both sides, we get:

$$\sin \theta = \pm \frac{\sqrt{3}}{2}$$

However, since $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$, we know that $\sin \theta$ is positive. Therefore, we can conclude that:

$$\sin \theta = \frac{\sqrt{3}}{2}$$

Q: What are some common applications of the Pythagorean identity?

A: The Pythagorean identity has many important applications in trigonometry, including:

• Solving trigonometric equations
• Finding the values of trigonometric functions
• Relating the sine, cosine, and tangent functions

Q: Can the Pythagorean identity be used to find the value of $\tan \theta$ given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$?

A: Yes, we can use the Pythagorean identity to find the value of $\tan \theta$ given that $\sin \theta = \frac{\sqrt{3}}{2}$ and $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$. First, we can square both sides of the equation $\sin \theta = \frac{\sqrt{3}}{2}$ to get:

$$\sin^2 \theta = \left(\frac{\sqrt{3}}{2}\right)^2$$

Simplifying the right-hand side, we get:

$$\sin^2 \theta = \frac{3}{4}$$

Now, we can substitute this value into the Pythagorean identity:

$$\frac{3}{4} + \cos^2 \theta = 1$$

Subtracting $\frac{3}{4}$ from both sides, we get:

$$\cos^2 \theta = \frac{1}{4}$$

Taking the square root of both sides, we get:

$$\cos \theta = \pm \frac{1}{2}$$

However, since $0^{\circ} \ \textless \ \theta \ \textless \ 90^{\circ}$, we know that $\cos \theta$ is positive. Therefore, we can conclude that:

$$\cos \theta = \frac{1}{2}$$

Now, we can use the definition of the tangent function to find the value of $\tan \theta$:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

Substituting the values of $\sin \theta$ and $\cos \theta$, we get:

$$\tan \theta = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}$$

Simplifying the right-hand side, we get:

$$\tan \theta = \sqrt{3}$$

Conclusion

In this article, we answered some of the most frequently asked questions about the Pythagorean identity and its applications. We showed how to use the Pythagorean identity to find the value of each given trigonometric function, including $\cos \theta$, $\sin \theta$, and $\tan \theta$. We hope that this article has provided a clear understanding of the Pythagorean identity and its applications.

Final Thoughts

The Pythagorean identity is a fundamental concept in trigonometry that has many important applications. It is used to solve trigonometric equations, find the values of trigonometric functions, and relate the sine, cosine, and tangent functions. We hope that this article has provided a clear understanding of the Pythagorean identity and its applications.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometry for Dummies" by Mary Jane Sterling

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry
• [3] Wolfram Alpha: Trigonometry